If S is a (multiplicative) Semigroup and a ∈ S, the binary operation ∘ defined on the set S by x ∘ y = x a y is associative and the resulting semigroup is called a variant of S. We study the congruence a defined on S by saying that two elements are α-related if and only if they determine the same variant of S. Certain quotients of variants are used to provide an arbitrary semigroup with a generalised local structure. The variant formulation of Nambooripad's partial order on a regular semigroup is used to show that the order possesses a certain property (involving D-equivalence).

In this paper, we prove that there exist two finite sets S1 (with 1 element) and S2 (with 3 elements) such that any two entire functions f and g satisfying Ef(Sj) = Eg(Sj) for j = 1, 2 must be identical. This answers a question posed by Gross. Examples are provided to show that this result is sharp.

A star is a connected graph in which every vertex but one has valency 1. This paper concerns the question of when complete graphs can be decomposed into stars, all of the same order, which have pairwise disjoint edge-sets. It is shown that the complete graphs on rm and rm + 1 vertices, r > 1, can be decomposed into stars with m edges, if and only if r is even or m is odd.

We give an upper bound for the number of zeros of recurrence sequences defined over an algebraic number field in terms of their order, the degree of their field of definition and the number of prime ideal divisors of the characteristic roots of the sequence.

The paper improves on an upper bound for the order of the Schur multiplier of a finite p-group given by Wiegold in 1969. The new bound is applied to the problem of classifying p-groups according to the size of their Schur multipliers.

We establish a correlation between the structures of a group of power automorphisms of some group and their mutual commutator subgroup and consider the consequences for the norm of a group, and for its capability.

Trace forms have been well studied as invariant quadratic forms on finite dimensional Lie algebras; the best known of these forms in the Cartan-Killing form. All those forms, however, have the ideal [L, L] ∩ R (with the radical R) in the orthogonal L⊥ and thus are frequently degenerate. In this note we discuss a general construction of Lie algebras equipped with non-degenerate quadratic forms which cannot be obtained by trace forms, and we propose a general structure theorem for Lie algebras supporting a non-degenerate invariant quadratic form. These results complement and extend recent developments of the theory of invariant quadratic forms on Lie algebras by Hilgert and Hofmann [2], keith [4], and Medina and Revoy [7].

A set of conditions for the almost sure convergence of a stochastic iterative procedure is given. The conditions are framed in terms of the behaviour of the random adjustment made the n-th step rather than in terms of some underlying regression model.

The general position number of a connected graph is the cardinality of a largest set of vertices such that no three pairwise-distinct vertices from the set lie on a common shortest path. In this paper it is proved that the general position number is additive on the Cartesian product of two trees.

Nelsen [J. Algebra 298 (2006) 134–141] asked whether there is a natural class of McCoy rings which includes all reversible rings and all rings R such that R[X] is semi-commutative. In this paper, some new equivalent conditions of McCoy rings are given. One of them is used to answer this question in the affirmative. Finally, an example is given which is McCoy and semi-commutative, but it is not reversible and does not have the property that R[x] is semi-commutative.

A square matrix is called cyclic if its characteristic and minimum polynomials coincide, separable if the characteristic polynomial has no repeated roots. Recent results of P. Neumann and Praeger, and of Lehrer, about the numbers of such matrices over a finite field are sharpened.

Using a dual variational method we shall show the existence of solutions to the Dirichlet problem without assuming Palais-Smale condition.

Some generalised invex conditions are given for a nondifferentiable constrained optimisation problem, generalising those of Hanson and Mond for differentiable problems. Some duality results are obtained.

We establish several theorems concerning properly embedded constant mean curvature surfaces (cmc-surfaces) in homogeneously regular 3-manifolds, when the mean curvature H is large.

A set of blocks which is a subset of a unique t – (v, k, λt) design is said to be a defining set of that design. We examine the properties of such a set, and show that its automorphism group is related to that of the whole design. Smallest defining sets are found for 2-designs and 3-designs on seven or eight varieties with block size three or four, revealing interesting combinatorial structures.

We give applications of the gauge-invariant uniqueness theorem, which states that the Cuntz-Krieger algebras of directed graphs are characterised by the existence of a canonical action of . We classify the C*-algebras of higher order graphs, identify the C*-algebras of cartesian product graphs with a certain fixed point algebra and investigate a relation called elementary shift equivalence on graphs and its effect on the associated graph C*-algebras.

If standard central difference formulas are used to compute second or third order derivatives from measured data even quite precise data can lead to totally unusable results due to the basic instability of the differentiation process. Here an averaging procedure is presented and analysed which allows the stable computation of low order derivatives from measured data. The new method first averages the data, then samples the averages and finally applies standard difference formulas. The size of the averaging set acts like a regularisation parameter and has to be chosen as a function of the grid size h.

A space of periodic generalised functions, called boehmians, is investigated. The space of boehmians contains all periodic distributions. It is known that not every hyperfunction is a boehmian. We show that the converse is also true. We present some theorems which give sufficient conditions for a sequence of complex numbers to be the Fourier coefficients of a boehmian. Sufficient conditions (in terms of the Fourier coefficients) are obtained for a sequence of boehmians to converge. As an application, a Dirichlet problem is discussed.

We show that nontrivial classical pretzel knots L (p, q, r) are hyperbolic with eight exceptions which are torus knots. We find Conway polynomials of n-pretzel links using a new computation tree. As applications, we compute the genera of n-pretzel links using these polynomials and find the basket number of pretzel links by showing that the genus and the canonical genus of a pretzel link are the same.

We consider relatively Meir–Keeler condensing operators to study the existence of best proximity points (pairs) by using the notion of measure of noncompactness, and extend a result of Aghajani et al. [‘Fixed point theorems for Meir–Keeler condensing operators via measure of noncompactness’, Acta Math. Sci. Ser. B35 (2015), 552–566]. As an application of our main result, we investigate the existence of an optimal solution for a system of integrodifferential equations.